Dynamic imaging may employ one of several medical imaging technologies, e.g., dynamic MRI, dynamic CT, dynamic optical imaging, and dynamic fluorescence, to acquire a time sequence of images. One of the primary uses of dynamic medical imaging is thus to examine motion, which carries with it the biological signature of malfunctions for various organs, especially the heart and the lungs. Dynamic imaging can also examine brain functions, exemplified by perfusion imaging.
Existing literature mainly employs two specific methodologies for dynamic imaging of organs such as the heart and the lungs. The first method relies on a set of static imaging acquisitions while the subject holds their breath, and resorts to image registration for retrieving the motion. The second method tries to capture free-breathing motion by using fast imaging techniques, with a trade-off between image resolution and quality.
FIG. 1A shows an example of an MRI imaging technique using traditional methods. Typically, an MRI image is obtained by sampling the object densely in Fourier space to obtain a k-space image 101. This acquired k-space image is then used to compute a real space image 103 by taking the inverse Fourier transform F* of the k-space data. However, for dynamic MRI applications, the time required to densely sample the k-space of the object is long, resulting in slow frame rates and motion induced errors. To help alleviate the above problems, the technique of compressive sensing (CS) has been used. Compressive sensing enables one to sample well below the Nyquist frequency while attaining accurate reconstruction of the original signal, under the assumption the signal is sparse in the basis under certain transforms, known as sparse transformations.
The CS process is shown graphically in FIG. 1B. In CS, the goal is to not densely sample the k-space when taking the initial image, but rather to sparsely sample the k-space as shown by the grey circles in partial k-space image 105. Rather than reconstructing the real space image using the inverse Fourier transform (a linear operator), CS relies on the existence of a non-linear sparse transform 109 (Ψ) to map the partial k-space image to a high resolution real space image 107. Because only a small fraction of the k-space is sampled during the sparse acquisition step, the time required to acquire a sparsely sampled k-space image 105 is less than the time required to acquire a densely sampled k-space image. Thus, CS is desirable for fast acquisition dynamic imaging. However, the success of CS depends on the existence of a suitable nonlinear transformation 109 for reconstructing a high-resolution real space image 107 from the sparsely sampled k-space image 105. Furthermore, the precise k-space measurement strategy used may contribute to the numerical robustness and/or computational difficulty of the reconstruction routine. The same may be said for CS strategies for dynamic imaging in real space, e.g., dynamic fluorescence imaging or the like.
Generally, CS may be summarized mathematically as follows. For dynamic imaging, e.g., images of a beating heart, a full resolution frame acquisition of the object would result in a sequence of full resolution video frames u(t). However, in CS u(t) is not directly measured. Rather, the CS problem may be posed as follows: given compressive measurement y(t) obtained by a series of CS measurements, recover the full resolution frames u(t) from y(t). In more mathematical terms, CS may be expressed by the following equations:
For real space: y(t)=Φ(t)·u(t)+e(t), where Φ(t) is the measurement matrix taking the value of any real number and e(t) is a noise term.
For Fourier space: y(t)=Φ(t)·F(u(t))+e(t) where Φ(t) is the measurement matrix taking the value of where {0,1} and e(t) is a noise term.
Thus, the goal of CS is to reconstruct u(t) using the knowledge of the measurement y(t) and the measurement matrix Φ(t). Note that for CS, the number of measurements (M) taken with each compressive acquisition is much less than the dimensionality (N) of the full-resolution image being sampled, i.e., M<<N. Stated another way, in CS the number of pixels in the CS frame is much less than the number of pixels in the full-resolution frame. Thus, with the use of CS techniques, an estimate to the full-resolution image u(t) may be reconstructed from an acquisition y(t) that includes only a compressive measurement of the full-resolution image.